ascPochhammer_nat_eq_natCast_descFactorial

English

For semiring S, and naturals a,b, ascPochhammer S b evaluated at a cast equals (a+b-1).descFactorial b cast.

Русский

Для полугруппы S и натуральных a,b: ascPochhammer S b-оценивается в a приводом равно (a+b−1).descFactorial b приводом.

LaTeX

$$$\\forall a,b:\\ (ascPochhammer\\ S\\ b).\\mathrm{eval}(a)^{cast} = (a+b-1).\\mathrm{descFactorial}\\ b^{cast}$$$

Lean4

theorem ascPochhammer_nat_eq_natCast_descFactorial (S : Type*) [Semiring S] (a b : ℕ) :
    (ascPochhammer S b).eval (a : S) = (a + b - 1).descFactorial b :=
  by
  norm_cast
  rw [ascPochhammer_nat_eq_descFactorial]

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